Led light through material
You need to do that in post. The compositor to be precise.

In the passes you need to enable the emission pass

In the compositor you can now see


Image
Alpha
Depth 
Emission


You add a glare node and set it to fog glow and plug the emission in the color inout, play with the settings till it matches the reference

Now you need a mix Color, plug the glare node in the top and the image node in the bottom.

Set the mix node to add

Now you can play with the values, the higher the add factor the more it will shine and look like it's scattering in the material and coming out glowing.

But rigth now it's not what you want, you want a scattering look. You need to add an object index value to the object and use this as a mask in the mix color node to only add the glow effect where the object is, this will make it seem like it goes throu the object and not throu the air too.

Note that this is the efficient variation. You can use subsurface scattering but it will pump your render times into infinity and beyond